cp's OEIS Frontend

A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).

%I A378379 #16 Dec 12 2024 23:08:39
%S A378379 1,1,2,3,4,6,7,9,10,12,14,16,18,20,23,25,28,30,33,35,38,41,44,47,50,
%T A378379 53,56,60,63,67,70,74,77,81,84,88,92,96,100,104,108,112,116,120,125,
%U A378379 129,134,138,143,147,152,156,161,165,170,175,180,185,190,195,200,205,210,215,220
%N A378379 Minimal x such that there is a partition of (x, x) into sums of distinct pairs of nonnegative integers with size at least n, excluding (0, 0).
%C A378379 For (n, n), there is at least one maximal partition P that's symmetric: (x, y) in P <=> (y, x) in P. This can be proven by manipulating integer sequences c(i) (i >= 1) such that 0 <= c(i) <= i+1 for all i and Sum_{i > 0} i*c(i) = 2n, which correspond to partitions P of (n, n) with size |P| = Sum_{i > 0} c(i), where c(i) is equal to number of (x, y) in P such that x+y = i.
%F A378379 a(n*(n+3)/2) = n*(n+1)*(n+2)/6.
%e A378379 For n = 8, a(n) = 9, as (9, 9) can be expressed as the sum (0, 1) + (0, 2) + (0, 3) + (1, 0) + (2, 0) + (3, 0) + (1, 2) + (2, 1), but the longest sum for (8, 8) has 7 pairs.
%o A378379 (Python)
%o A378379 import math
%o A378379 def A378379(n: int) -> int:
%o A378379   l = (math.isqrt(1+8*n)-1)//2 # l = A003056(n), min. possible largest pair norm
%o A378379   r = n - (l-1)*(l+2)//2 # r = n - A000096(l-1), number of pairs with norm l
%o A378379   return ((l-1)*l*(l+1)//3 + l*r + 1)//2 # ceil((A007290(l+1) + l*r) / 2)
%Y A378379 Maximal size among partitions considered by A054242 and A201377.
%Y A378379 Minimal x such that A378126(x, x) >= n.
%Y A378379 Cf. A086435.
%K A378379 nonn
%O A378379 1,3
%A A378379 _Jimin Park_, Nov 24 2024